Abstract
The paper considers the development of an analytical-numerical method for solving the Dirichlet–Neumann problem for the equation \(\operatorname{div} (\varkappa \nabla u) = 0\) in a domain \(\Omega\) containing a cone with base of general form, including a polyhedral corner. The coefficient \(\varkappa\) of the equation is piecewise constant with discontinuity along a conical surface lying in \(\Omega\) and having the vertex in common with the original cone. The solution of the problem is represented as the limit of a sequence of linear combinations of functions \(\Psi_k\) that make up an approximation system and are constructed explicitly. The method permits one to obtain not only the solution in the domain \(\Omega\) but also its expansion near the vertex of the cone and to calculate the corresponding singularity exponents and intensity factors. Some numerical results are presented.
